1. Corporate Finance Portfolio Theory Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES

2. A.Farber Vietnam 2004 |2|2 Portfolio selection Objectives for this session –1. Gain a better understanding of the rational for benefit of diversification –2. Identify measures of systematic risk : covariance and beta –3. Analyse the choice of an optimal portfolio

3. A.Farber Vietnam 2004 |3|3 Combining the Riskless Asset and a single Risky Asset Consider the following portfolio P: Fraction invested –in the riskless asset 1-x (40%) –in the risky asset x (60%) Expected return on portfolio P: Standard deviation of portfolio : Riskless asset Risky asset Expected return 6%12% Standard deviation 0%20%

4. A.Farber Vietnam 2004 |4|4 Relationship between expected return and risk Combining the expressions obtained for : the expected return the standard deviation leads to

5. A.Farber Vietnam 2004 |5|5 Risk aversion Risk aversion : For a given risk, investor prefers more expected return For a given expected return, investor prefers less risk Expected return Risk Indifference curve

6. A.Farber Vietnam 2004 |6|6 Utility function Mathematical representation of preferences a: risk aversion coefficient u = certainty equivalent risk-free rate Example: a = 2 A 6% 0 0.06 B 10% 10% 0.08 = 0.10 - 2×(0.10)² C 15% 20% 0.07 = 0.15 - 2×(0.20)² B is preferred Utility

7. A.Farber Vietnam 2004 |7|7 Optimal choice with a single risky asset Risk-free asset : R F Proportion = 1-x Risky portfolio S: Proportion = x Utility: Optimum: Solution: Example: a = 2

8. A.Farber Vietnam 2004 |8|8 Diversification

9. A.Farber Vietnam 2004 |9|9 A measure of systematic risk : beta Consider the following linear model R t Realized return on a security during period t  A constant : a return that the stock will realize in any period R Mt Realized return on the market as a whole during period t  A measure of the response of the return on the security to the return on the market u t A return specific to the security for period t (idosyncratic return or unsystematic return)- a random variable with mean 0 Partition of yearly return into: –Market related part ß R Mt –Company specific part  + u t

10. A.Farber Vietnam 2004 | 10 Beta - illustration Example: Suppose Rt = 2% + 1.2 RMt + ut If RMt = 10% The expected return on the security given the return on the market E[Rt |RMt] = 2% + 1.2 x 10% = 14% If Rt = 17%, ut = 17%-14% = 3%

11. A.Farber Vietnam 2004 | 11 Covariance and correlation Statistical measures of the degree to which random variables move together Covariance Like variance figure, the covariance is in squared deviation units. Not too friendly... Correlation covariance divided by product of standard deviations Covariance and correlation have the same sign –Positive : variables are positively correlated –Zero : variables are independant –Negative : variables are negatively correlated The correlation is always between –1 and + 1

12. A.Farber Vietnam 2004 | 12 Risk and expected returns for porfolios In order to better understand the driving force explaining the benefits from diversification, let us consider a portfolio of two stocks (A,B) Characteristics: –Expected returns : –Standard deviations : –Covariance : Portfolio: defined by fractions invested in each stock X A, X B X A + X B = 1 Expected return on portfolio: Variance of the portfolio's return:

13. A.Farber Vietnam 2004 | 13 Example Invest $ 100 m in two stocks: A $ 60 m X A = 0.6 B $ 40 m X B = 0.4 Characteristics (% per year) A B Expected return 20% 15% Standard deviation 30% 20% Correlation 0.5 Expected return = 0.6 × 20% + 0.4 × 15% = 18% Variance = (0.6)²(.30)² + (0.4)²(.20)²+2(0.6)(0.4)(0.30)(0.20)(0.5)  ²p = 0.0532  Standard deviation = 23.07 % Less than the average of individual standard deviations: 0.6 x0.30 + 0.4 x 0.20 = 26%

14. A.Farber Vietnam 2004 | 14 Diversification effect Let us vary the correlation coefficient Correlation coefficient Expected return Standard deviation -1 18 10.00 -0.5 18 15.62 0 18 19.7 0.5 18 23.07 1 18 26.00 Conclusion: –As long as the correlation coefficient is less than one, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities

15. A.Farber Vietnam 2004 | 15 The efficient set for two assets: correlation = +1

16. A.Farber Vietnam 2004 | 16 The efficient set for two assets: correlation = -1

17. A.Farber Vietnam 2004 | 17 The efficient set for two assets: correlation = 0

18. A.Farber Vietnam 2004 | 18 Marginal contribution to risk: some math Consider portfolio M. What happens if the fraction invested in stock I changes? Consider a fraction X invested in stock i Take first derivative with respect to X: Value the derivative for X = 0

19. A.Farber Vietnam 2004 | 19 Marginal contribution to risk: conclusion Risk of portfolio increase if and only if: The marginal contribution of stock i to the risk is

20. A.Farber Vietnam 2004 | 20 Choosing portfolios from many stocks Porfolio composition : (X 1, X 2,..., X i,..., X N ) X 1 + X 2 +... + X i +... + X N = 1 Expected return: Risk: Note: N terms for variances N(N-1) terms for covariances Covariances dominate

21. A.Farber Vietnam 2004 | 21 Example Consider the risk of an equally weighted portfolio of N "identical« stocks: Equally weighted: Variance of portfolio: If we increase the number of securities ?: Variance of portfolio:

22. A.Farber Vietnam 2004 | 22 Conclusion 1. Diversification pays - adding securities to the portfolio decreases risk. This is because securities are not perfectly positively correlated 2. There is a limit to the benefit of diversification : the risk of the portfolio can't be less than the average covariance (cov) between the stocks The variance of a security's return can be broken down in the following way: The proper definition of the risk of an individual security in a portfolio M is the covariance of the security with the portfolio: Total risk of individual security Portfolio risk Unsystematic or diversifiable risk